Rigidity of some functional inequalities on RCD spaces

Abstract

We study the cases of equality and prove a rigidity theorem concerning the 1-Bakry-Émery inequality. As an application, we prove the rigidity of the Gaussian isoperimetric inequality, the logarithmic Sobolev inequality and the Poincaré inequality in the setting of $\mathrm{RCD}(K,\infty )$ metric measure spaces. This unifies and extends to the non-smooth setting the results of Carlen-Kerce, Morgan, Bouyrie, Ohta-Takatsu, Cheng-Zhou. Examples of non-smooth spaces fitting our setting are measured-Gromov Hausdorff limits of Riemannian manifolds with uniform Ricci curvature lower bound, and Alexandrov spaces with curvature lower bound. Some results including the rigidity of $\Phi$-entropy inequalities, the rigidity of the 1-Bakry-Émery inequality are of independent interest even in the smooth setting.

References in corpus (9)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure
• The splitting theorem in non-smooth context
• Well posedness of Lagrangian flows and continuity equations in metric measure spaces
• Local Poincaré inequalities from stable curvature conditions on metric spaces
• Rigidity for the spectral gap on $RCD(K,\infty )$-spaces
• Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds
• Rigidity phenomenons for an infinite dimension diffusion operator and cases of near equality in the Bakry--Ledoux isoperimetric comparison Theorem